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(a) Check that {1, 2} is an orthogonal set with the weight function w(x) = x2...
Chapter 2. Legendre Polynomials Examples Show that each function set is orthogonal in the given interval...
Chapter 2. Legendre Polynomials Examples Show that each function set is orthogonal in the given interval with respect to the specified weight function a. {sin mx}, (-7,7], w(x) = 1 b. {1, 2, 3 (3x2 - 1)}, [-1, 1], w(x) = 1 c. {1, 1 – 2, 3 (x2 - 4x + 2)}; (0,00), w(x) = e-6 Theorem: If the set of functions {P(x)} is orthogonal, then any piece-wise contin- uous function in [a, b] can be represented by the...
Please write neat and show work/steps 3. Consider the function f(x) = (4x +5 on the...
Please write neat and show work/steps
3. Consider the function f(x) = (4x +5 on the interval (-1.1). (a) Find the quadratic Taylor approximation fr(x) > 00 + 10 + c2x2. Calculate the C to four decimal places. (b) Find the quadratic Legendre approximation f1(x) -- 20 +ajx + a2x?. Calculate the a; to four decimal places. If the two approximations differ greatly, something is probably wrong. You may want to consult section 4 in the pdf I sent you...
Verify by direct integration that the functions are orthogonal with respect to the indicated weight function...
Verify by direct integration that the functions are orthogonal with respect to the indicated weight function w(x) on the given interval. 4p(x) = 1, 4,6x) = -x + 1, 12(X) = 2*2 - 2x - 2x + 1; w(x) - e*, [0, 0) Using integration by parts we find the following. (In the last step of each integral, simplify your answer completely.) 6 *wcx360(87%)/(x) dx = 60 1) ox 11-62 6o *wcxXq6x)22() dx = 6* 1) ox 11 + 1*2*x+...
e interval -1,1].if f.ge C[L.] 7 The field of play is C the space of all functions that are continuous on th we'...
e interval -1,1].if f.ge C[L.] 7 The field of play is C the space of all functions that are continuous on th we'll define the inner product as (f.g)= 5(x)g(x)dx. The question is simply this: Find the orthogonal projection of e onto P, and graph both functions on [-2,2].
e interval -1,1].if f.ge C[L.] 7 The field of play is C the space of all functions that are continuous on th we'll define the inner product as (f.g)= 5(x)g(x)dx. The...
2. Consider the vector space C([0, 1]) consisting of all continuous functions f: [0,1]-R with the...
2. Consider the vector space C([0, 1]) consisting of all continuous functions f: [0,1]-R with the weighted inner product, (f.g)-f(x) g(x) x dr. (a) Let Po(z) = 1, Pi(z) = x-2, and P2(x) = x2-6r + 흡 Show that {Po, pi,r) are orthogonal with respect to this inner product b) Use Gram-Schmidt on f(x)3 to find a polynomial pa(r) which is orthogonal to each of po P1 P2 You may use your favorite web site or software to calculate the...
1 0 < x < 1, 2. Consider the Haar scaling function, p(x):= { +, and...
1 0 < x < 1, 2. Consider the Haar scaling function, p(x):= { +, and (x) := { -1 10 otherwise 0 0 < x < 1/2, 1/2 < x < 1,. Sketch (by hand is okay) 7(2x) and 7(2x – 1). Show these functions form an orthogonal set. Find the otherwise corresponding orthonormal set. in 3. Let V be a vector space with a complex inner product (,) > Suppose that the set S= {U1, U2, ..., Un}...
2. (a) Prove that 1 (f, g)=| x2 f(x)g(x)dx is an inner product on the vector space C(I-1,1) of co...
NEED (B) AND (C)
2. (a) Prove that 1 (f, g)=| x2 f(x)g(x)dx is an inner product on the vector space C(I-1,1) of continuous real-valued funo- tions on the domain [-1, 1] (b) Use the Gram-Schmidt process to find an orthonormal basis for P2(R) with re- spect to this inner product (c) Find a polynomial q(x) such that for every p E P2R
2. (a) Prove that 1 (f, g)=| x2 f(x)g(x)dx is an inner product on the vector space...
Determine if each basis is orthogonal. Further, is the basis orthonormal? (a) In the vector space...
Determine if each basis is orthogonal. Further, is the basis orthonormal? (a) In the vector space R3 (i.e. column vectors in 3-space): -1 1 ( 2 5 3 -3 (b) In the vector space that consists of polynomial functions of degree less than or equal to 2: {f(x) = x2 – 3, g(x) = 4, h(x) = x2 +2} (c) In the vector space that consists of 2x2 matrices: (You'd decided what the inner product was on a previous math...
6. Determine if each basis is orthogonal. Further, is the basis orthonormal? (a) In the vector...
6. Determine if each basis is orthogonal. Further, is the basis orthonormal? (a) In the vector space R3 (i.e. column vectors in 3-space): 3 -1 2 3 1 5 -3 (b) In the vector space that consists of polynomial functions of degree less than or equal to 2: {f(x) = x2 – 3, 9(x) = 4, h(x) = x2 +2} (c) In the vector space that consists of 2 x 2 matrices: (You'd decided what the inner product was on...
Let W Span((2,-3,0, 1), (4,-6,-2, 1), (6,-9,-2,2) R4. (a) Find a basis for W (b) Find a basis for...
Let W Span((2,-3,0, 1), (4,-6,-2, 1), (6,-9,-2,2) R4. (a) Find a basis for W (b) Find a basis for W (c) Find an orthogonal basis for W and W (d) The union of these two orthogonal bases (put the basis for W and W what? Why is the union orthogonal? into one set) is an orthogonal basis for
Let W Span((2,-3,0, 1), (4,-6,-2, 1), (6,-9,-2,2) R4. (a) Find a basis for W (b) Find a basis for W (c) Find...
Chapter 2. Legendre Polynomials Examples Show that each function set is orthogonal in the given interval with respect to the specified weight function a. {sin mx}, (-7,7], w(x) = 1 b. {1, 2, 3 (3x2 - 1)}, [-1, 1], w(x) = 1 c. {1, 1 – 2, 3 (x2 - 4x + 2)}; (0,00), w(x) = e-6 Theorem: If the set of functions {P(x)} is orthogonal, then any piece-wise contin- uous function in [a, b] can be represented by the...
Please write neat and show work/steps
3. Consider the function f(x) = (4x +5 on the interval (-1.1). (a) Find the quadratic Taylor approximation fr(x) > 00 + 10 + c2x2. Calculate the C to four decimal places. (b) Find the quadratic Legendre approximation f1(x) -- 20 +ajx + a2x?. Calculate the a; to four decimal places. If the two approximations differ greatly, something is probably wrong. You may want to consult section 4 in the pdf I sent you...
Verify by direct integration that the functions are orthogonal with respect to the indicated weight function w(x) on the given interval. 4p(x) = 1, 4,6x) = -x + 1, 12(X) = 2*2 - 2x - 2x + 1; w(x) - e*, [0, 0) Using integration by parts we find the following. (In the last step of each integral, simplify your answer completely.) 6 *wcx360(87%)/(x) dx = 60 1) ox 11-62 6o *wcxXq6x)22() dx = 6* 1) ox 11 + 1*2*x+...
e interval -1,1].if f.ge C[L.] 7 The field of play is C the space of all functions that are continuous on th we'll define the inner product as (f.g)= 5(x)g(x)dx. The question is simply this: Find the orthogonal projection of e onto P, and graph both functions on [-2,2].
e interval -1,1].if f.ge C[L.] 7 The field of play is C the space of all functions that are continuous on th we'll define the inner product as (f.g)= 5(x)g(x)dx. The...
2. Consider the vector space C([0, 1]) consisting of all continuous functions f: [0,1]-R with the weighted inner product, (f.g)-f(x) g(x) x dr. (a) Let Po(z) = 1, Pi(z) = x-2, and P2(x) = x2-6r + 흡 Show that {Po, pi,r) are orthogonal with respect to this inner product b) Use Gram-Schmidt on f(x)3 to find a polynomial pa(r) which is orthogonal to each of po P1 P2 You may use your favorite web site or software to calculate the...
1 0 < x < 1, 2. Consider the Haar scaling function, p(x):= { +, and (x) := { -1 10 otherwise 0 0 < x < 1/2, 1/2 < x < 1,. Sketch (by hand is okay) 7(2x) and 7(2x – 1). Show these functions form an orthogonal set. Find the otherwise corresponding orthonormal set. in 3. Let V be a vector space with a complex inner product (,) > Suppose that the set S= {U1, U2, ..., Un}...
NEED (B) AND (C)
2. (a) Prove that 1 (f, g)=| x2 f(x)g(x)dx is an inner product on the vector space C(I-1,1) of continuous real-valued funo- tions on the domain [-1, 1] (b) Use the Gram-Schmidt process to find an orthonormal basis for P2(R) with re- spect to this inner product (c) Find a polynomial q(x) such that for every p E P2R
2. (a) Prove that 1 (f, g)=| x2 f(x)g(x)dx is an inner product on the vector space...
Determine if each basis is orthogonal. Further, is the basis orthonormal? (a) In the vector space R3 (i.e. column vectors in 3-space): -1 1 ( 2 5 3 -3 (b) In the vector space that consists of polynomial functions of degree less than or equal to 2: {f(x) = x2 – 3, g(x) = 4, h(x) = x2 +2} (c) In the vector space that consists of 2x2 matrices: (You'd decided what the inner product was on a previous math...
6. Determine if each basis is orthogonal. Further, is the basis orthonormal? (a) In the vector space R3 (i.e. column vectors in 3-space): 3 -1 2 3 1 5 -3 (b) In the vector space that consists of polynomial functions of degree less than or equal to 2: {f(x) = x2 – 3, 9(x) = 4, h(x) = x2 +2} (c) In the vector space that consists of 2 x 2 matrices: (You'd decided what the inner product was on...
Let W Span((2,-3,0, 1), (4,-6,-2, 1), (6,-9,-2,2) R4. (a) Find a basis for W (b) Find a basis for W (c) Find an orthogonal basis for W and W (d) The union of these two orthogonal bases (put the basis for W and W what? Why is the union orthogonal? into one set) is an orthogonal basis for
Let W Span((2,-3,0, 1), (4,-6,-2, 1), (6,-9,-2,2) R4. (a) Find a basis for W (b) Find a basis for W (c) Find...
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